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The next-to-top term in knot Floer homology 结花同源性的下一个最上面的术语
IF 1.1 2区 数学
Quantum Topology Pub Date : 2021-04-29 DOI: 10.4171/qt/174
Yi Ni
{"title":"The next-to-top term in knot Floer homology","authors":"Yi Ni","doi":"10.4171/qt/174","DOIUrl":"https://doi.org/10.4171/qt/174","url":null,"abstract":"Let $K$ be a null-homologous knot in a generalized L-space $Z$ with $b_1(Z)le1$. Let $F$ be a Seifert surface of $K$ with genus $g$. We show that if $widehat{HFK}(Z,K,[F],g)$ is supported in a single $mathbb Z/2mathbb Z$--grading, then [mathrm{rank}widehat{HFK}(Z,K,[F],g-1)gemathrm{rank}widehat{HFK}(Z,K,[F],g).]","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"19 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79137376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Non-formality in PIN(2)-monopole Floer homology PIN(2)中的非正式性-单极花同源性
IF 1.1 2区 数学
Quantum Topology Pub Date : 2021-03-26 DOI: 10.4171/QT/151
Francesco Lin
{"title":"Non-formality in PIN(2)-monopole Floer homology","authors":"Francesco Lin","doi":"10.4171/QT/151","DOIUrl":"https://doi.org/10.4171/QT/151","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"27 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2021-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80331455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Instanton Floer homology, sutures, and Euler characteristics 瞬花同源性、缝合线和欧拉特性
IF 1.1 2区 数学
Quantum Topology Pub Date : 2021-01-13 DOI: 10.4171/qt/182
Zhenkun Li, Fan Ye
{"title":"Instanton Floer homology, sutures, and Euler characteristics","authors":"Zhenkun Li, Fan Ye","doi":"10.4171/qt/182","DOIUrl":"https://doi.org/10.4171/qt/182","url":null,"abstract":"This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler characteristic $chi_{rm gr}$ of this homology is fully determined by the axioms we proposed. As a result, we conclude that $chi_{rm gr}(SHI(M,gamma))=chi_{rm gr}(SFH(M,gamma))$ for any balanced sutured manifold $(M,gamma)$. In particular, for any link $L$ in $S^3$, the Euler characteristic $chi_{rm gr}(KHI(S^3,L))$ recovers the multi-variable Alexander polynomial of $L$, which generalizes the knot case. Combined with the authors' earlier work, we provide more examples of $(1,1)$-knots in lens spaces whose $KHI$ and $widehat{HFK}$ have the same dimension. Moreover, for a rationally null-homologous knot in a closed oriented 3-manifold $Y$, we construct canonical $mathbb{Z}_2$-gradings on $KHI(Y,K)$, the decomposition of $I^sharp(Y)$ discussed in the previous paper, and the minus version of instanton knot homology $underline{rm KHI}^-(Y,K)$ introduced by the first author.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2021-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86984989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
Quantum invariants of three-manifolds obtained by surgeries along torus knots 三流形沿环面结整形的量子不变量
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-11-11 DOI: 10.4171/qt/175
H. Murakami, Anh T. Tran
{"title":"Quantum invariants of three-manifolds obtained by surgeries along torus knots","authors":"H. Murakami, Anh T. Tran","doi":"10.4171/qt/175","DOIUrl":"https://doi.org/10.4171/qt/175","url":null,"abstract":"We study the asymptotic behavior of the Witten-Reshetikhin-Turaev invariant associated with the square of the $n$-th root of unity with odd $n$ for a Seifert fibered space obtained by an integral Dehn surgery along a torus knot. We show that it can be described as a sum of the Chern-Simons invariants and the twisted Reidemeister torsions both associated with representations of the fundamental group to the two-dimensional complex special linear group.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"21 6 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90409169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Web calculus and tilting modules in type $C_2$ Web演算和倾斜模块类型$C_2$
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-09-29 DOI: 10.4171/qt/166
Elijah Bodish
{"title":"Web calculus and tilting modules in type $C_2$","authors":"Elijah Bodish","doi":"10.4171/qt/166","DOIUrl":"https://doi.org/10.4171/qt/166","url":null,"abstract":"Using Kuperberg's $B_2/C_2$ webs, and following Elias and Libedinsky, we describe a \"light leaves\" algorithm to construct a basis of morphisms between arbitrary tensor products of fundamental representations for $mathfrak{so}_5cong mathfrak{sp}_4$ (and the associated quantum group). Our argument has very little dependence on the base field. As a result, we prove that when $[2]_qne 0$, the Karoubi envelope of the $C_2$ web category is equivalent to the category of tilting modules for the divided powers quantum group $mathcal{U}_q^{mathbb{Z}}(mathfrak{sp}_4)$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"64 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74067365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
A closed formula for the evaluation of foams 评价泡沫的封闭公式
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-08-22 DOI: 10.4171/qt/139
Louis-Hadrien Robert, E. Wagner
{"title":"A closed formula for the evaluation of foams","authors":"Louis-Hadrien Robert, E. Wagner","doi":"10.4171/qt/139","DOIUrl":"https://doi.org/10.4171/qt/139","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"129 1","pages":"411-487"},"PeriodicalIF":1.1,"publicationDate":"2020-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77029449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 29
Non-semisimple 3-manifold invariants derived from the Kauffman bracket 由Kauffman括号导出的非半简单三流形不变量
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-07-21 DOI: 10.4171/QT/164
M. Renzi, J. Murakami
{"title":"Non-semisimple 3-manifold invariants derived from the Kauffman bracket","authors":"M. Renzi, J. Murakami","doi":"10.4171/QT/164","DOIUrl":"https://doi.org/10.4171/QT/164","url":null,"abstract":"We recover the family of non-semisimple quantum invariants of closed oriented 3-manifolds associated with the small quantum group of $mathfrak{sl}_2$ using purely combinatorial methods based on Temperley-Lieb algebras and Kauffman bracket polynomials. These invariants can be understood as a first-order extension of Witten-Reshetikhin-Turaev invariants, which can be reformulated following our approach in the case of rational homology spheres.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"30 17 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84731137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Tensor product categorifications, Verma modules and the blob 2-category 张量积分类，Verma模和blob 2类
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-05-13 DOI: 10.4171/QT/156
Abel Lacabanne, Gr'egoire Naisse, P. Vaz
{"title":"Tensor product categorifications, Verma modules and the blob 2-category","authors":"Abel Lacabanne, Gr'egoire Naisse, P. Vaz","doi":"10.4171/QT/156","DOIUrl":"https://doi.org/10.4171/QT/156","url":null,"abstract":"We construct a dg-enhancement of Webster's tensor product algebras that categorifies the tensor product of a universal sl2 Verma module and several integrable irreducible modules. We show that the blob algebra acts via endofunctors on derived categories of such dg-enhanced algebras in the case when the integrable modules are two-dimensional. This action intertwines with the categorical action of sl2. From the above we derive a categorification of the blob algebra.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"53 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87478265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Holonomy invariants of links and nonabelian Reidemeister torsion 连杆的完整不变量与非abel Reidemeister扭转
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-05-03 DOI: 10.4171/QT/160
Calvin McPhail-Snyder
{"title":"Holonomy invariants of links and nonabelian Reidemeister torsion","authors":"Calvin McPhail-Snyder","doi":"10.4171/QT/160","DOIUrl":"https://doi.org/10.4171/QT/160","url":null,"abstract":"We show that the reduced $mathrm{SL}_2(mathbb{C})$-twisted Burau representation can be obtained from the quantum group $mathcal{U}_q(mathfrak{sl}_2)$ for $q = i$ a fourth root of unity and that representations of $mathcal{U}_q(mathfrak{sl}_2)$ satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the $operatorname{SL}_2(mathbb{C})$-twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet, Geer, Patureau-Mirand, and Reshetikhin, and we interpret their invariant as a twisted Conway potential.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"10 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88601129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
A unification of the ADO and colored Jones polynomials of a knot 一个结的ADO和彩色琼斯多项式的统一
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-03-22 DOI: 10.4171/qt/161
Sonny Willetts
{"title":"A unification of the ADO and colored Jones polynomials of a knot","authors":"Sonny Willetts","doi":"10.4171/qt/161","DOIUrl":"https://doi.org/10.4171/qt/161","url":null,"abstract":"In this paper we prove that the family of colored Jones polynomials of a knot in $S^3$ determines the family of ADO polynomials of this knot. More precisely, we construct a two variables knot invariant unifying both the ADO and the colored Jones polynomials. On one hand, the first variable $q$ can be evaluated at $2r$ roots of unity with $r in Bbb N^*$ and we obtain the ADO polynomial over the Alexander polynomial. On the other hand, the second variable $A$ evaluated at $A=q^n$ gives the colored Jones polynomials. From this, we exhibit a map sending, for any knot, the family of colored Jones polynomials to the family of ADO polynomials. As a direct application of this fact, we will prove that every ADO polynomial is q-holonomic and is annihilated by the same polynomials as of the colored Jones function. The construction of the unified invariant will use completions of rings and algebra. We will also show how to recover our invariant from Habiro's quantum $mathfrak{sl}_2$ completion studied in arXiv:math/0605313.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"175 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79746984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 16
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